The set of quantities Tcb{\displaystyle {\mathfrak {T}}_{c}^{b}} might be called the stress-energy-complex (comp. § 38). As for a change of the system of coordinates the transformation formulae for T{\displaystyle {\mathfrak {T}}} are similar to those by which tensors are defined, we can also speak of the stress-energy-tensor. We have namely

for all variations δgab{\displaystyle \delta g_{ab}} which vanish at the boundary of the field of integration together with their first derivatives. The index ψ{\displaystyle \psi } in the first term indicates that in the variation of L{\displaystyle \mathrm {L} } the quantities ψab{\displaystyle \psi _{ab}} must be kept constant.

If we suppose L{\displaystyle \mathrm {L} } to be expressed in the quantities gab{\displaystyle g^{ab}} and if (42), (45) and (48) are taken into consideration, we find from (61) that at each point of the field-figure

↑The notations ψab,ψab¯{\displaystyle \psi _{ab},{\overline {\psi _{ab}}}} and ψab∗{\displaystyle \psi _{ab}^{*}} (see (27), (29) and § 11, 1915), will however be preserved though they do not correspond to those of Einstein. As to formulae (59) and (60) it is to be understood that if p{\displaystyle p} and q{\displaystyle q} are two of the numbers 1, 2, 3, 4, p′{\displaystyle p'} and q′{\displaystyle q'} denote the other two in such a way that the order pqp′q′{\displaystyle p\ q\ p'\ q'} is obtained from 1 2 3 4 by an even number of permutations of two ciphers.
If x1,x2,x3,x4{\displaystyle x_{1},x_{2},x_{3},x_{4}} are replaced by x,y,z,t{\displaystyle x,y,z,t} and if for the stresses the usual notations Xx,Xy{\displaystyle X_{x},X_{y}}, etc., are used (so that e.g. for a surface element dσ{\displaystyle d\sigma } perpendicular to the axis of x,Xx{\displaystyle x,X_{x}} is the first component of the force per unit of surface which the part of the system situated on the positive side of dτ{\displaystyle d\tau } exerts on the opposite part) then T11=Xx,T12=Xy{\displaystyle {\mathfrak {T}}_{1}^{1}=X_{x},{\mathfrak {T}}_{1}^{2}=X_{y}}, etc. Further −T14,−T24,−T34{\displaystyle -{\mathfrak {T}}_{1}^{4},-{\mathfrak {T}}_{2}^{4},-{\mathfrak {T}}_{3}^{4}} are the components of the momentum per unit of volume and T41,T42,T43{\displaystyle {\mathfrak {T}}_{4}^{1},{\mathfrak {T}}_{4}^{2},{\mathfrak {T}}_{4}^{3}} the components of the energy-current. Finally T44{\displaystyle {\mathfrak {T}}_{4}^{4}} is the energy per unit of volume.